论文标题
签名的热带半空间和凸度
Signed tropical halfspaces and convexity
论文作者
论文摘要
我们将热带凸度的基本面扩展到了向热带正面的矫正，扩大了Loho andVégh（ITCS 2020）开发的理论。我们研究了两个称为“ to-convexity”（以前是“签名的热带凸度”）和新颖的概念“ TC-Convexity”的热带数字的凸性概念。我们得出了几个分离结果，以实现codvexity和Tc-convexity。一个关键的成分是对TC-Hemispaces的透彻理解 - 那些补体也是TC-Convex的TC-Convex集。此外，我们在Puiseux系列的凸度和其签名估值之间的相互作用中使用了新的见解。值得注意的是，TC-跨性别性可以看作是代表定向的矩阵的自然凸概念，因为它是由载体在定向的矩阵中的组成操作的概括而产生的。我们通过以TC-Convexity在真实热带超场上的线性空间表示线性空间来显式。
We extend the fundamentals for tropical convexity beyond the tropically positive orthant expanding the theory developed by Loho and Végh (ITCS 2020). We study two notions of convexity for signed tropical numbers called 'TO-convexity' (formerly 'signed tropical convexity') and the novel notion 'TC-convexity'. We derive several separation results for TO-convexity and TC-convexity. A key ingredient is a thorough understanding of TC-hemispaces - those TC-convex sets whose complement is also TC-convex. Furthermore, we use new insights in the interplay between convexity over Puiseux series and its signed valuation. Remarkably, TC-convexity can be seen as a natural convexity notion for representing oriented matroids as it arises from a generalization of the composition operation of vectors in an oriented matroid. We make this explicit by giving representations of linear spaces over the real tropical hyperfield in terms of TC-convexity.
